On the Spectra of the Gravity Water Waves Linearized at Monotone Shear Flows

We consider the spectra of the 2-dim gravity waves of finite depth linearized at a uniform monotonic shear flow \(U(x_2)\), \(x_2 \in (-h, 0)\), where the wave numbers k of the horizontal variable \(x_1\) is treated as a parameter. Our main results include a.) a complete branch of non-singular neutral modes \(c^+(k)\) strictly decreasing in \(k\ge 0\) and converging to U(0) as \(k \rightarrow \infty \); b.) another branch of non-singular neutral modes \(c_-(k)\), \(k \in (-k_-, k_-)\) for some \(k_->0\), with \(c_-(\pm k_-) = U(-h)\); c.) the non-degeneracy and the bifurcation at \((k_-, c=U(-h))\); d.) the existence and non-existence of unstable modes for c near U(0), \(U(-h)\), and interior inflection values of U; e.) the complete spectral distribution in the case where \(U''\) does not change sign or changes sign exactly once non-degenerately. In particular, U is spectrally stable if \(U'U''\le 0\) and unstable if U has a non-degenerate interior inflection value or \(\{U'U''>0\}\) accumulate at \(x_2=-h\) or 0. Moreover, if U is an unstable shear flow of the fixed boundary problem in a channel, then strong gravity could cause instability of the linearized gravity waves in all long waves (i.e. \(|k|\ll 1\)).